Professor Luc Vinet is one of Canada leading mathematical and theoretical physicists who has made outstanding contributions in numerous areas. The unifying feature of his research is the innovative use of group theoretical and algebraic methods, the emphasis on exact solutions of physical problems and the originality of his approach. He has made important contributions that have had great impact on both physics and mathematics. He explored various algebraic structures appropriate to describe symmetries in different physical problems. These go well beyond standard Lie groups and algebras. They include polynomial, quantum, super- and para super-algebras. He is very well known for his influential work on quantum many body problems and for his application of this work to a proof of the long outstanding Macdonald conjecture on properties of multivariate orthogonal polynomials. His contributions to the symmetry theory of difference and q-difference equations are truly pioneering. In 2011, he has discovered new families of orthogonal polynomials, associated to reflections. These have already found many applications. In the context of quantum information theory, he has shown how spin chains can be used to design perfect quantum wires.

Lecture Abstract

It is known that perfect state transfer (PST) can be achieved in XX spin chains with properly engineered nearest-neighbour (NN) couplings. The simplest model with this feature is based on the Krawtchouk polynomials. In view of the mathematical equivalence between the equations governing the dynamics of single excitations in XX spin chains and those describing photon propagation in arrays of evanescently coupled waveguides, PST can be experimentally realized in photonic lattices. In this context restricting to NN interactions is obviously an approximation. The phenomenon of fractional revival (FR) or wave packet splitting that has smaller but identical packets reproduce periodically, can also be seen in certain XX spin chains. This is not so however in the NN Krawtchouk model. Like PST, FR brings useful new tools in quantum information and can generate for instance quantum entanglement. I shall present an analytic extension of the NN Krawtchouk model that includes next-tonearest neighbour couplings. Under certain conditions, it will be shown to admit PST as well as FR in distinction to the NN situation. Its application to coherent transport in photonic lattices will be discussed.